For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, angles.

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2
votes
2answers
43 views

Quadrilateral's area problem

I have some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$, $N$, $P$ and $Q$ are the midpoints of the sides $AB$, $BC$, $CD$ and $AD$. $AN$, $BP$, $MD$ and $CQ$ are ...
1
vote
1answer
17 views

Distance between $2$ skew lines

Suppose that $A(0,0,0), B(1,2,0), C(0,-3,2)$ and $D(3,-4,5)$ and $AB, AC$ and $AD$ are three edges of a parallelepiped. If $l_1$ is the line passing through $A$ and $B$ and $l_2$ is the line passing ...
0
votes
1answer
79 views

Geometry - angle bisector, circumcircle: SL olympiad

I tried this problem as much as I can, but I got nothing. This is a Sri Lankan mathematical olympiad problem. Let $P$,$Q$ be points on the sides $AB$ and $AC$, respectively, of a $\triangle ABC$ ...
2
votes
0answers
23 views

Collinearity problem (Newton-Gauss line)

I had some troubles with this problem : Let $ABCD$ be a convex quadrilateral. $M$ and $N$ are the midpoints of the diagonals $AC$ and $BD$. The sides $AB$ and $CD$ are extended until they ...
3
votes
3answers
42 views

$\cos \alpha+\cos(\alpha+\beta)+\cos(\alpha+2\beta)+…+\cos(\alpha+(n-1)\beta)=0 $

If each side of a regular polygon of $n$ sides subtend an angle $\alpha$ at the center of the polygon and each exterior angle of the polygon is $\beta$,then prove that $\cos ...
0
votes
0answers
16 views

Locus of points on a rotating line ; points differently ordered

A line rotates about a fixed point $O$ with ordered points $P,O,M $, while $ M $ is moving along this line $POM$. Find locus of points $ P ,M $ if $ MP^2- OM^2 = T^2 $ constant for all inclinations ...
2
votes
0answers
48 views

Intersecting convex figures

Take in the real plane a finitely long horizontal line segment and connect the two endpoints by a convex path, above the segment, with the property that the only extreme points of the convex hull of ...
0
votes
2answers
11 views

Rectangle Zoom-in: Accounting for proportions and distances between them

Let's say I have n rectangles: each with their own height and width, and each with their own coordinate on a plane. I can scale the width and height of the rectangles by let's say...S. How do I ...
2
votes
2answers
27 views

About finding maximum area [closed]

What is the maximum area of a triangle if two vertices are given?as I need to find the no. Of possibilities of third vertices ? I will then find the possibilities of third vertex using given area if ...
-3
votes
2answers
30 views

Reflection of a curve(advanced) [closed]

x=logy (base 10) is the reflection of y=logx (base 10) about the line whose equation is? An answer with a detailed approach is appreciated. Thanks in advance. By the way, the answer is y=x
1
vote
2answers
23 views

Disk in $\mathbb R^2$ with uniform norm

I am having a trouble understanding a definition. The points are in $\mathbb R^2$, and the author defines $\delta(p, r)$ to be an $l_\infty$ disk of radius $r$ centred at $p$. I just learned what the ...
1
vote
0answers
46 views

Practical application of Gauss-Lucas theorem

Let $z_1,z_2,z_3 \in \mathbb C$ pairwise distinct be the affix of points $A, B$ and $C$. Let $P(x)=(x-z_1)(x-z_2)(x-z_3)$. Let $z_4$ and $z_5$ be the roots of $P'$ (with the possibilty that ...
0
votes
1answer
24 views

constructing segments with equal cross ratio

I was puzzeling again and had the following problem: Given: two segments $AD$ and $PS$ on $AD$ there are points $B$ and $C$ so that $AD \gt AC \gt AB$ (so they are in order A, B , C, D ) on $PS$ ...
1
vote
1answer
40 views

Prove area of a quadrilateral is $\frac14[4m^2n^2-(b^2+d^2-a^2-c^2)^2]^{\frac12}$

Someone asked me this question which I am really stuck at, any help is appreciated. If $a,b,c,d$ are the sides of a quadrilateral and $m,n$ are diagonals of the quadrilateral, then prove that ...
-5
votes
0answers
37 views

Angle of rolled cone [closed]

EDIT 5: The problem concerns application of geodesics on a cone. A rectangular sheet of flat paper [ Sides B1,L1,B2,L2 ] is folded/rolled forming into a right circular cone of semi-vertical ...
2
votes
1answer
19 views

Ceiling height required for standing upright a piece of furtniture

Say you have a piece of furniture with height $h$ and depth $d$ that you want to stand upright from the floor, what is the ceiling height required $H$? (Apoligies if the graphic isn't correct, the ...
0
votes
1answer
19 views

Determination of directional deformation of deformed and shifted 2D object (triangle)

I have an engineering problem. Let's assume there is a 2D structure that undergoes deformation. None of the points stay in initial location and the structure (imagining a triangle should suffice) can ...
7
votes
1answer
225 views

The case of Captain America's shield: a variation of Alhazen's Billard problem

I'm sure a lot of you are acquainted with Alhazen's Billiard problem, which involves finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in ...
-2
votes
0answers
24 views

How to find coordinates of points on a 2D surface embedded in 3D space

kindly assist with this problem. Given an equilateral triangle in 2D plane (see figure 1) with origin (0,0) at point B, the coordinates of points A and C can be calculated as A(acos60,asin60) and ...
0
votes
1answer
42 views

Prove continuity of a function that is defined through a geometric construction

I need to prove that a function is continuous, but it is not defined explicitly,it's like this: given a point $P$ on a circumference and an angle $0\le a\le \frac{\pi}{2}$ defined by $P$ and another ...
0
votes
1answer
29 views

Mind refresher on a few simple algebra-geometry problems

I feel silly for asking this, but I've completely forgotten some steps on how to do a few of these simple algebra/geometry problems. 1) Simplify $\sqrt{18x}-4\sqrt{x^3}$. I rearranged this to ...
3
votes
0answers
45 views

Proof that 10 lines pass through the centroid of a triangle

Let $A$, $B$, $C$, $D$, and $E$ be points on a circle. For any three points, we draw the line going through the centroid of the triangle formed by these three points that is perpendicular to the line ...
1
vote
2answers
33 views

How to express $\phi$ in terms of $R\text{, }x\text{ and }\theta$

Let $S$ be a circle with radius $R$ and center at $O$. Let $P$ be any arbitrary point inside circle such that its distance from $O$ is $x$ and the ray $\overrightarrow{OP}$ cuts the circle $S$ at ...
1
vote
4answers
39 views

Find the equation $ax + by + cz = d$ of the plane which has equal distance to the points $A(1, 2, 3)$ and $B(4, 5, 6)$

I was just wondering if anyone has any suggestions as to how to compute this equation? Find the equation $ax + by + cz = d$ of the plane for which every point has equal distance to the points ...
2
votes
2answers
168 views

Given 3 spheres, find the equation of the plane that touches each of the spheres on the same side..?

I have a problem I am trying to solve, but I have no idea how to solve it. If I have 3 spheres, $A(1, 2, 0), B(4, 5, 0), \text{ and } C(1, 3, 2)$ of radius 1, how would I go about finding the ...
-1
votes
2answers
31 views

Given two rectangles A and B and their dimensions, is a there test for lengthA<lengthB, widthA<widthB [closed]

If we have two rectangles and their dimensions, is there a mathematical test to simultaneously compare the two numbers. For examples, if we wanted to compare the area, we would compare the products, ...
4
votes
2answers
56 views

If $a_1,a_2,a_3,…,a_n$ are the side lengths of $A_1A_2A_3…A_n$ convex polygon,then$\frac{a^2_1+a^2_2+a^2_3+…+a^2_{n-1}}{a^2_n}$ is

If $a_1,a_2,a_3,...,a_n$ are the side lengths of $A_1A_2A_3...A_n$ convex polygon,then$\frac{a^2_1+a^2_2+a^2_3+....+a^2_{n-1}}{a^2_n}$ is ...
2
votes
1answer
18 views

Determine whether the locus of the point P will intersect the straight line $y=-1$

A point $P(x,y)$ moves in such a way that its distance from the point $A(3,1)$ is always three times its distance from the straight line $x=-1$. (a) Find the equation of the locus of the moving point ...
7
votes
1answer
60 views

What is maximum a number of to form right-triangles from in n straight lines

I am interested what is maximum a number of to form right-triangles from in $n=100$ straight lines such $n=3$,then maximum number of is $1$,see fig:$\Delta ABC$ is right-triangles. $n=4$ then ...
0
votes
1answer
26 views

Prove synthetically: projection of a circle onto a plane is an ellipse

I am wondering how I can prove synthetically that the projection of a circle onto a plane is an ellipse.
0
votes
2answers
27 views

Domain Technicality

If $f(x)=x^2$ and $g(x)=\frac{1}{x+1},$ then what is the domain of $h(x)=\frac{f(x)}{g(x)}$? I know that the resultant function is $h(x)=x^3+x^2,$ but is there a hole in the graph of $h(x)$ at ...
3
votes
2answers
45 views

A plane contains a set of marked points, such that any three can be covered by a unit disk. Prove that the entire set can be covered by a unit disk.

A set of points is marked on the plane, with the property that any three marked points can be covered with a disk of radius 1. Prove that the set of all marked points can be covered with a disk ...
0
votes
0answers
18 views

Maximizing the area of a transformed rectangle within some bounds

I need to solve this problem for a program I'm writing, but I'm struggling a bit with the maths behind it. Given a rectangle $R_{max\_layout}$ and a $3\times 3$ transformation matrix $M_{transform}$, ...
2
votes
1answer
47 views

definition of trapezoid

In my country, for many years, trapezoid is defined as such in the textbooks: a quadrilateral with only two parallel sides. But today, referring to foreign sources, someone told me that: a ...
2
votes
2answers
43 views

Finding a perpendicular vector from a line to a point

Let's say I have an arbitrary linear line described by $y=mx+b$. I also have a point $P(x_1,y_1)$ that is not on that line. I suppose $P$ can be anywhere relative to the line. How can I find the ...
-2
votes
1answer
85 views

Locus of points on a rotating line

A line rotates about a fixed point $O$ with ordered points O,P,and M moving along the line. Find locus of points P and M if $ OM^2 - MP^2 = T^2 $ constant for all inclinations of $OP$. The ...
1
vote
1answer
49 views

How to prove a lemma required for the Banach Tarski Paradox?

I tried to teach myself the proof of the Banach Tarski Paradox by reading Terence Tao's paper on the subject; the link to the paper is here: ...
1
vote
2answers
34 views

Determine coordinates for Mandelbrot set zoom.

I am writing a computer program to produce a zoom on the Mandelbrot set. The issue I am having with this is that I don't know how to tell the computer where to zoom. As of right now I just pick a ...
0
votes
0answers
32 views

Parametrization of the $Ax=b, x \geq 0$ domain for Monte-Carlo simulation

I have a linear system, $n=15$, with $6$ constraints. There's no problem finding a single solution or establishing the null space; so I can see the full solution space. But I'm only interested in ...
2
votes
2answers
35 views

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.

The sum of the lengths of the hypotenuse and another side of a right angled triangle is given.The area of the triangle will be maximum if the angle between them is: ...
9
votes
2answers
80 views

Finding tangents to a circle with a straightedge

There is a geometric construction that I heard years ago and I still haven't figured out why it works despite several attempts. Playing with pen, paper and GeoGebra makes me confident that it does ...
1
vote
0answers
11 views

Prove that $\{m \in S_{\sigma} \, | \, \gamma(m) \neq 0\}$ is a face of $\sigma^V \cap M$

I am trying to solve exercise 3.2.6 pag.124 of Cox, Little, Schenck book http://www.math.colostate.edu/~renzo/teaching/Toric14/CoxLittleShenck.pdf because it is required to prove orbit-cone ...
1
vote
1answer
35 views

Calculating the volume of a surfboard

I'm building a website for a client in which customers can customise the shape of their board (curvature, length, width, thickness, and so forth) and the client has asked if we can calculate the ...
7
votes
2answers
264 views

A box contained within larger box has a smaller surface area than the larger box?

Suppose we have a box (parallelepiped) A completely contained within another box B. Is the surface area of A nessecarily less than the surface area of the B? Edit: note that the sides of A are not ...
1
vote
1answer
22 views

Upper bound for area on sphere

Consider the sphere $\mathbb{S}^{n-1}:= \{x \in \mathbb{R}^n : \|x\|_2=1\}$, and let $A^\epsilon_x:= \{z \in \mathbb{S}^{n-1}:\langle z,x \rangle \ge \epsilon\}$ where $x \in \mathbb{S}^{n-1}$. Note ...
0
votes
0answers
15 views

Derivation in eulerian angles

In the following link page 393 of the book http://uqu.edu.sa/files2/tiny_mce/plugins/filemanager/files/4320535/Analytical_Mechanics_Fowles_7ed__English.pdf I don't understand at all how did they ...
0
votes
0answers
21 views

Winning the relay race for your team

Relay race, members of a team of three take turns running from the point P to a point on the circle; To A for the first, B for the second, and C for the third, starting and returning to point P, ...
0
votes
1answer
32 views

Constructing points in a triangle

[OH] is an altitude segment in triangle MOR which is right at O. Let D be a point on [OM] and E be a point of [OR] such that [OH] passes through the midpoint of [DE]. The question is to justify the ...
2
votes
2answers
168 views

On pentagonal tilings

The following image has been in the news recently: My understanding is that these are all the known (to-date) tilings of the plane using convex pentagons. Can someone explain to me why the ...
1
vote
2answers
16 views

Flatland analogy of a hypersphere passing through our space

So, I realize this is really hard to convey without diagrams, but I find the wikipedia representation of a hypersphere really lacking. The stereographic projection shows "infinite radius" meridians ...